A292865 Determinants of the symmetric matrices whose entries on and below the main diagonal correspond to those of Pascal's triangle.

1, 0, -1, 8, -71, 656, -4816, 1920, 168784, 43920880, -3315147449, 209095006856, -19095123359744, 1814464114046976, 320005209305667584, -253215321875947192320, -3298397219599339984896, 24417272707694829159671808, 265094852554176756050442657024, -931723550682987095264656018072440

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = det(L_n+U_n-I_{n+1}), where L_n is the lower triangular Pascal matrix of order n, U_n is the transpose of L_n and I_n is the identity matrix of order n. Note that L_n, U_n and I_n all have determinant 1 for all n.

Example

a(0) is the determinant of the 1 X 1 matrix whose sole entry is one.
a(1) is the determinant of the 2 X 2 matrix of all ones.
a(2) is the determinant of the 3 X 3 matrix
[1 1 1]
[1 1 2]
[1 2 1].
a(3) is the determinant of the 4 X 4 matrix
[1 1 1 1]
[1 1 2 3]
[1 2 1 3]
[1 3 3 1].

Mathematica

PascalMatrix = Function[n, Table[Table[Binomial[m, i], {i, 0, n}], {m, 0, n}]];
PascalDet = Function[n, Det[PascalMatrix[n] + Transpose[PascalMatrix[n]] - IdentityMatrix[n + 1]]];
Table[PascalDet[i], {i, 0, 19}]

Prog

(Python)
from sympy import *
def m(N):
    return Matrix([
        ([binomial(i, n) for n in range(i+1)] +[0] * (N-i))
        for i in range(N+1)
    ])
def matrix(N):
    return m(N) + m(N).transpose() - eye(N+1)
[ matrix(i).det() for i in range(20)]
# Gilles Castel, Sep 25 2017

Crossrefs

Sequence in context: A070998 A370178 A187709 * A152265 A081178 A096341

Adjacent sequences: A292862 A292863 A292864 * A292866 A292867 A292868

Keyword

sign

Author

Alexander Farrugia, Sep 25 2017

Status

approved